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Home , Urination

Duration of does not change with body size

Patricia J. Yanga, Jonathan Phama, Jerome Chooa, and David L. Hua,b,1

Schools of aMechanical Engineering and bBiology, Georgia Institute of , Atlanta, GA 30332

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 14, 2014 (received for review February 6, 2014)

Many urological studies rely on models of animals, such as rats and generate jets. Instead, they urinate using a series of drops, which pigs, but their relation to the human is poorly is shown by the 0.03-kg lesser -faced fruit bat and the 0.3-kg understood. Here, we elucidate the hydrodynamics of urination rat in Fig. 2 A–C, respectively. across five orders of magnitude in body mass. Using high-speed Fig. 1H shows the urination time for 32 animals across six videography and flow-rate measurement obtained at Zoo Atlanta, orders of magnitude of body mass from 0.03 to 8,000 kg. De- we discover that all mammals above 3 kg in weight empty their spite this wide range in mass, urination time remains constant, = ± = bladders over nearly constant duration of 21 ± 13 s. This feat is T 21 13 s (n 32), for all animals heavier than 3 kg. This ’ possible, because larger animals have longer and thus, invariance is noteworthy, considering that an elephant sblad- ’ higher gravitational force and higher flow speed. Smaller mam- der, at 18 L, is nearly 3,600 times larger in volume than a s mals are challenged during urination by high viscous and capillary bladder at 5 mL. Using the method of least squares, we fit the ∼ 0.13 forces that limit their to single drops. Our findings reveal data to a clear scaling law shown by the dashed line: T M that the is a flow-enhancing device, enabling the urinary (Fig. 1H). system to be scaled up by a factor of 3,600 in volume without For small animals, urination is a high-speed event of 0.01- to 2-s duration and therefore, quite different from the behavior of compromising its function. This study may help to diagnose uri- the large animals observed that urinate for 21 s. Fig. 1H shows nary problems in animals as well as inspire the design of scalable urination time across 11 small animals, including one bat, five hydrodynamic systems based on those in nature. rats, and five mice. Their body masses ranged from 0.03 to 0.3 kg. The large error bar for the rats is caused by bladder fullness ’ | allometry | scaling | Bernoulli s principle varying across individuals. Fig. 2D shows the time course of the urine drop’s radius measured by image analysis of high-speed edical and veterinary urology often relies on simple, non- video of a rat. To rationalize the striking differences between Minvasive methods to characterize the health of the urinary large and small animals, we turn to mathematical modeling of system (1, 2). One of the most easily measured characteristics of the urinary system. the urinary system is its flow rate (3), changes in which may be used to diagnose urinary problems. The expanding of Modeling Assumptions. Urination may be simply described math- aging males may constrict the urethra, decreasing urine flow ematically. Fig. 1E shows a schematic of the urinary system, rate (4). Obesity may increase abdominal pressure, causing consisting of a bladder of volume V and the urethra, which is incontinence (5). Studies of these ailments and others often assumed to be a straight vertical pipe of length L and diameter involve animal subjects of a range of sizes (6). A study of D. We assume that the urethra has such a thin wall that its in- ternal and external diameters are equal. Urination begins when urination in zero gravity involved a rat suspended on two legs for the smooth muscles of the bladder pressurize the urine to P , long periods of time (7), whereas other studies involve mice (8), bladder measured relative to atmospheric pressure. After an initial tran- (1), and pigs (9). Despite the wide range of animals used in sient of duration that depends on the system size, a steady flow of urological studies, the consequences of body size on urination speed u is generated. remain poorly understood. Previous medical and veterinary studies, particularly cystome- The bladder serves a number of functions, as reviewed by trography and ultrasonography, report substantial data on the Bentley (10). In desert animals, the bladder stores water to be anatomy, pressure, and flow rate of the urinary system. Fig. 3 retrieved at a time of need. In mammals, the bladder acts as shows urethral length (8, 15–25) and diameter (15, 24–34), flow a waterproof reservoir to be emptied at a time of convenience. rate (35–42), bladder capacity (25, 43–49), and bladder pressure This control of urine enables animals to keep their homes sanitary (1, 35, 39, 40, 43, 46, 50) for over 100 individuals across 13 species. and themselves inconspicuous to predators. Stored urine may even be used in defense, which one knows from handling Significance and pets. Several misconceptions in urology have important repercus- sions in the interpretation of healthy bladder function. For in- Animals eject fluids for waste elimination, communication, and stance, several investigators state that urinary flow is driven defense from predators. These diverse systems all rely on the entirely by bladder pressure. Consequently, their modeling of the fundamental principles of fluid mechanics, which we use to bladder neglects gravitational forces (11–13). Others, such as predict urination duration across a wide range of mammals. In Martin and Hillman (14), contend that urinary flow is driven by this study, we report a mathematical model that clarifies mis- a combination of both gravity and bladder pressure. In this study, conceptions in urology and unifies the results from 41 in- we elucidate the hydrodynamics of urination across animal size, dependent urological and anatomical studies. The theoretical showing the effects of gravity increase with increasing body size. framework presented may be extended to study fluid ejection from animals, a universal phenomenon that has received little Results attention. In Vivo Experiments. We filmed the urination of 16 animals and obtained 28 videos of urination from YouTube, listed in SI Ap- Author contributions: P.J.Y. and D.L.H. designed research; J.P. and J.C. performed re- pendix. Movies S1–S4 show that urination style is strongly de- search; P.J.Y. and D.L.H. analyzed data; and P.J.Y. and D.L.H. wrote the paper. pendent on animal size. Here, we define an animal as large if it is The authors declare no conflict of interest. heavier than 3 kg and an animal as small if it is lighter than 1 kg. This article is a PNAS Direct Submission. Large animals, from dogs to elephants, produce jets and sheets 1To whom correspondence should be addressed. Email: [email protected]. – of urine, which are shown in Fig. 1 A D. Small animals, including This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. rodents, bats, and juveniles of many mammalian species, cannot 1073/pnas.1402289111/-/DCSupplemental.

11932–11937 | PNAS | August 19, 2014 | vol. 111 | no. 33 www.pnas.org/cgi/doi/10.1073/pnas.1402289111 Downloaded by guest on September 25, 2021 SCIENCES APPLIED PHYSICAL SCIENCES APPLIED BIOLOGICAL

Fig. 1. Jetting urination by large animals, including (A) elephant, (B) cow, (C) goat, and (D) dog. Inset of cow is reprinted from the public domain and cited in SI Appendix.(E) Schematic of the urinary system. (F) Ultrasound image of the bladder and urethra of a female human. The straight arrow indicates the urethra, and the curved arrow indicates the bladder. Reproduced with permission from ref. 20, (Copyright 2005, Radiological Society of North America). (G) Transverse histological sections of the urethra from a female pig. Reproduced with permission from ref. 9, (Copyright 2001, Elsevier). (H) The relationship between body mass and urination time.

Table 1 shows the corresponding allometric relationships to be This shape factor is nearly constant across species and body used in numerical predictions for flow rate and urination time. mass and consistent with the value of 0.17 found by Wheeler We begin by showing that the urinary system is isometric (i.e., et al. (55). it has constant proportions across animal size). Fig. 3A shows Peak bladder pressure is difficult to measure in vivo, and in- the relation between body mass M and urethral dimensions stead, it is estimated using pressure transducers placed within the (length L and diameter D). As shown by the nearly parallel bladders of anesthetized animals. Pressure is measured when the trends for L and D (L = 35M0.43 and D = 2M0.39), the aspect bladder is filled to capacity by the injection of fluid. This tech- ratio of the urethra is 18. Moreover, the exponents are close to nique yields a nearly constant bladder pressure across animal 1/3 = ± = the expected isometric scaling of M . Fig. 3B shows the re- size: Pbladder 5.2 0.86 kPa (n 8), which is shown in Fig. 3D. lationship between body mass and bladder capacity. The bladder’s The constancy of bladder pressure at 5.2 kPa is consistent with capacity is V ∼ M0.97, and the exponent of near unity indicates other systems in the body. One prominent example is the re- isometry. spiratory system, which generates pressures of 10 kPa for animals In ultrasonic imaging (Fig. 1F), the urethra appears circular spanning from a mosquito to an elephant (56). (20). However, in histology (Fig. 1G), the urethra is clearly corrugated, which decreases its cross-sectional area (9). The Steady-State Equation of Urine Flow. We model the flow as steady state and the urine as an incompressible fluid of density ρ, presence of such corrugation has been verified in studies in μ σ which flow is driven through the urethra (51, 52), although the viscosity , and surface tension . The energy equation re- precise shape has been too difficult to measure. We proceed lates the pressures involved, each of which has units of energy by using image analysis to measure cross-sectional area A per cross-sectional area of the urethra per unit length down from urethral histological diagrams of dead animals in the the urethra: absenceofflow(9,53,54).Wedefineashapefactorα = 4A/ 2 πD , which relates the urethral cross-sectional area with re- Pbladder + Pgravity = Pinertia + Pviscosity + Pcapillary: [1] specttothatofacylinderofdiameterD.Fig.3C shows the shape factor α = 0.2 ± 0.05 (n = 5) for which the corrected Each term in Eq. 1 may be written simply by considering the cross-sectional area is 20% of the original area considered. pressure difference between the entrance and exit of the

Yang et al. PNAS | August 19, 2014 | vol. 111 | no. 33 | 11933 Downloaded by guest on September 25, 2021 Fig. 2. Dripping urination by small animals. (A)A rat’s excreted urine drop. (B) A urine drop released by the lesser dog-faced fruit bat Cynopterus bra- chyotis. Courtesy of Kenny Breuer and Sharon Swartz. (C) A rat’s urine drop grows with time. Inset is reprinted from the public domain and cited in SI Appendix.(D) Time course of the drop radii of the rat (triangles) along with prediction from our model (blue dotted line, α = 0.5; green solid line, α = 1; red dashed line, α = 0.2).

urethra. The combination of bladder and hydrostatic pressure our derivations here, however, we assume that the transient drives urine flow. Bladder pressure Pbladder is a constant given phase is negligible. in Fig. 3D. We do not model the time-varying height in the bladder, because bladders vary greatly in shape (57). Thus, Large Animals Urinate for Constant Duration. Bladder pressure, hydrostatic pressure scales with urethral length: Pgravity ∼ ρgL, gravity, and inertia are dominant for large animals, which can be where g is the acceleration of gravity. Dynamic pressure Pinertia shown by consideration of the dimensionless groups in SI Ap- scales as ρu2/2 and is associated with the inertia of the flow. pendix. Eq. 2 reduces to The viscous pressure drop in a long cylindrical pipe is given by the Darcy–Weisbach equation (58): P = f (Re)ρLu2/2αD. ρu2 pffiffiffi viscosity D P + ρgL = : [4] We use αD as the effective diameter of the pipe to keep the cross- bladder 2 sectional area of the pipe consistent with experiments. The Darcy friction factor fD is a function of the Reynolds number The urination time T, the time to completely empty the bladder, Re = ρuD/μ,suchthatfD(Re) = 64/Re for laminar flow and may be written as the ratio of bladder capacity to time-averaged = 1/4 4 < < 5 flow rate, T = V/Q. We define the flow rate as Q = uA, where A = fD(Re) 0.316/Re for turbulent flow (10 Repffiffiffi 10 ). Drops απD2/4 is the cross-sectional area of the urethra. Using Eq. 4 to generated from an orifice of effectivepffiffiffi diameter αD experience substitute for flow speed yields a capillary force (59) of Pcapillary = 4σ= αD. Substituting these terms into Eq. 1, we arrive at 4V =   : [5] T 1=2 ρu2 ρLu2 4σ απD2 2Pbladder + 2gL P + ρgL = + f ðReÞ + pffiffiffi : [2] ρ bladder 2 D 2αD αD By isometry, bladder capacity V ∼ M and urethral length and The relative magnitudes of the five pressures enumerated in 1/3 2 diameter both scale with M ; substitution of these scalings into Eq. are prescribed by six dimensionless groups, including the Eq. 5 yields urination time T ∼ M1/6 ≈ M0.16 in the limit of in- aforementioned Reynolds numberpffiffiffiffiffiffi and Darcy friction factor = = = ρ 2 σ creasing body mass. Agreement between predicted and mea- and well-known Froude Fr u gL and Bond Bo gD / sured scaling exponents is very good (0.13 compared with 0.16). numbers (60) as well as dimensionless groups pertaining to the We, thus, conclude that our scaling has captured the observed urinary system, the urethra aspect ratio As = D/L, and pressure = ρ invariance in urination time. ratio Pb Pbladder/ gL. Using these definitions, we nondimen- We go beyond a simple scaling by substituting the measured sionalize Eq. 2 to arrive at allometric relationships from Table 1 for L, D, α, V, and Pbladder into Eq. 5, yielding a numerical prediction for urination time. 2 2 This prediction (Fig. 1H, solid line) is shown compared with ex- + = Fr + ð Þ Fr + p4Asffiffiffi : [3] Pb 1 fD Re perimental values (Fig. 1H, dashed line). The general trend is 2 2αAs αBo captured quite well. We note that numerical values are under- predicted by a factor of three across animal masses, likely because In the following sections, we solve Eq. 3 in the limits of large and of the angle and cross-section of the urethra in vivo. small animals. How can an elephant empty its bladder as quickly as a cat? In SI Appendix, we apply a variation of the Washburn law Larger animals have longer urethras and therefore, greater hy- (61) to show that the steady-state model given in Eq. 2 is drostatic pressure driving flow. Greater pressures lead to higher accurate for most animals. Animals lighter than 100 kg ach- flow rates, enabling the substantial bladders of larger animals to ieve 90% of their flow velocity within 4 s; however, for animals be emptied in the same duration as those of their much smaller such as elephants, the transient phase can be substantial. For counterparts.

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Fig. 3. The relation between body mass M and properties of the urinary system. (A) Urethral length L (green triangles) and diameter D (blue circles). (B) SCIENCES

Bladder capacity V.(C) Shape factor α associated with the urethral cross-section. (D) Bladder pressure Pbladder.(E)Flowrateofmales.(F) Flow rate of females. APPLIED PHYSICAL Symbols represent experimental measurements, dashed lines represent best fits to the data, and solid lines represent predictions from our model.

Our model provides a consistent physical picture on consid- pressures and urethral anatomy (15, 16, 50) (Pbladder = 6.03 kPa, eration of flow rate. Combining Eq. 4 and the definition of flow L = 20 mm, D = 0.8 mm). rate (Q = uA) yields To determine urination time, we turn to the dynamics of drop

2 SCIENCES   filling. A spherical drop is filled by the influx of urine, Q = απD u/4. απ 2 1=2 D 2Pbladder By conservation of mass, dVdrop/dt = Q, a first-order differential Q = + 2gL : [6] APPLIED BIOLOGICAL 4 ρ equation that may be easily integrated to obtain the drop volume V(t). We assume that the initial droppffiffiffi corresponds to a sphere of the Our model gives insight into the distinct flow-rate scalings same diameter as the urethra, αD.Thus,theradiusofthe observed for both male and female mammals. Male mammals spherical urine drop may be written generally stand on four legs and have a that extends ! 1=3 3 3 2 downward, enabling them to urinate vertically. Assuming isometry α2D 3αD ut ∼ 1/3 ∼ 1/3 ∼ 5/6 ≈ 0.83 RðtÞ = + : [8] (D M and L M ), flow rate scales as Q M M 8 16 in the limit of large body mass. This predicted exponent is with- ∼ 0.92 in 10% of the observed scaling for males: QM M .Bysub- 8 7 stituting the allometric relations from Table 1 into Eq. 6,we Combining Eq. and the numerical solution for Eq. , we com- compute a numerical prediction for flow rate (Fig. 3E, solid line) pute the time course of the drop radius. This prediction is com- that is five times higher than experimental flow rates (Fig. 3E, pared with experimental values in Fig. 2D. We find that the prediction is highly sensitive to the value of α. Without consid- dashed line). This overprediction is roughly consistent with our α = underprediction for urination time. eration of the corrugated cross-section, a prediction of 1 Female mammals have anatomy such that the urethral exit is (Fig. 2D, green solid line) yields a flow rate that is too high. α = near the : thus, many female animals urinate horizontally. Using the shape factor 0.2 (Fig. 2D, red dashed line), our 2/3 = The scaling of Eq. 6 without the gravitational term is Q ∼ M ≈ model predicts a flow speed of u 1.2 m/s, which fits the data M0.67, and the exponent is in correspondence to that found in our fairly well. Using nonlinear least-squares fitting in Matlab, the 0.66 experiments for females: QF ∼ M . Substituting allometric best fit to the experimental data yields an intermediate value of relations from Table 1 yields a numerical prediction (Fig. 3F, α = 0.5 (Fig. 2D, blue dotted line). solid line) that remains in good agreement with experiments. The drop does not grow without limit but falls when its gravi- π 3ρ = tational force, scaling as 4 Rf g 3, overcomespffiffiffi its attaching cap- Small Animals Urinate Quickly and for Constant Duration. Bladder illary force to the urethra, scaling as π αDσ. Equating these two pressure, viscous pressure, and capillary pressure are dominant forces yields the final drop radius before detachment, for small animals, which is shown by the associated dimension- less groups in SI Appendix. In this limit, Eq. 2 reduces to  pffiffiffi  σ α 1=3 = 3 D ; [9] ρu2 32μLu 4σ Rf ρ P = + + pffiffiffi ; [7] 4 g bladder 2 αD2 αD which does a fair job of predicting the drop size. We predict drop which we solve numerically for flow speed u. To predict the flow radii for rats and mice of 1.3 and 1.1 mm, respectively, which are speed of a rat, inputs to this equation include the rat’s bladder two times as large as experimental values of 2.0 ± 0.1 (n = 5) and

Yang et al. PNAS | August 19, 2014 | vol. 111 | no. 33 | 11935 Downloaded by guest on September 25, 2021 Table 1. Measured allometric relationships for the urinary urethra. In the medical literature, the function of the urethra was system of animals previously unknown. It was simply defined as a conduit between Variable Unit Best fit R2 N bladder and genitals. In this study, we find that the urethra is analogous to Pascal’s Barrel: by providing a water-tight pipe to Duration of urination T s 8.2 M0.13 0.2 32 direct urine downward, the urethra increases the gravitational Urethral length L mm 35 M0.43 0.9 47 force acting on urine and therefore, the rate at which urine is Urethral diameter D mm 2.0 M0.39 0.9 22 expelled from the body. Thus, the urethra is critical to the Shape factor α — 0.2 M−0.05 0.5 5 bladder’s ability to empty quickly as the system is scaled up. Bladder capacity V mL 4.6 M0.97 0.9 9 Engineers may apply this result to design a system of pipes and −0.01 Bladder pressure Pbladder kPa 5.2 M 0.02 8 reservoirs for which the drainage time does not depend on system 0.66 Flow rate for females QF mL/s 1.8 M 0.9 16 size. This concept of a scalable hydrodynamic system may be used 0.92 Flow rate for males QM mL/s 0.3 M 0.9 15 in the design of portable reservoirs, such as water towers, water backpacks, and storage tanks. Body mass M given in kilograms. Duration of urination corresponds to animals heavier than 3 kg. Urethral length and diameter, shape factor, blad- Why is urination time 21 s, and why is this time constant across der capacity, bladder pressure, and flow rates correspond to animals heavier animal sizes? The numerical value of 21 s arises from the un- than 0.02 kg. derlying physics involving the physical properties of urine as well as the dimensions of the urinary system. Our model shows that differences in bladder capacity are offset by differences in flow 2.2 ± 0.4 mm (n = 5), respectively. We suspect this difference is rate, resulting in a bladder emptying time that does not change caused by our underestimation of urethral perimeter at the exit. with system size. Such invariance has been observed in a number For such a large drop to remain attached, we require the attach- of other systems. Examples include the height of a jump (64) and ment diameter to be larger by a factor of two, which is quite the number of heartbeats in a lifetime (65). Many of these examples possible, because the urethral exit is elliptical. arise from some aspect of isometry, such as with our system. Substituting Eq. 9 into Eq. 8, the time to eject one drop may From a biological perspective, the invariance of urination time be written suggests its low functional significance. Because bladder volume pffiffiffi ! is 4.6 mL/kg body mass and daily urine voided is 26 mL/kg body 16 αD 3 cos θ 1 4D cos θ 1 T = − ≈ pffiffiffi : [10] mass (66), mammals urinate 5.6 times/d. Because the time to drop 3 α 3u 4α2Bo 8 Bo u urinate once is 21 s, the daily urination time is 2 min, which can be translated to 0.2% of an animal’s day, a negligible portion The predictions of maximum drop size and time to fall are in compared with other daily activities, such as and sleeping, excellent correspondence with observed values for rats and mice. for which most animals take care to avoid predation. Thus, uri- Using Eq. 10, we predict drop falling times of 0.05 and 0.15 s for rats nation time likely does not influence animal fitness. The geometry and mice, respectively, which are nearly identical to experimental of the urethra, however, may play a role in other activities of values of 0.06 ± 0.05 (n = 5) and 0.14 ± 0.1 s (n = 14), respectively. high functional significance, such as ejaculation. A scaling for urine duration for small animals is not straight- In our study, we found that urination time is highly sensitive forward because of the nonlinearity of Eq. 7. We conduct a scaling to urethral cross-section. This dependency is particularly high for analysis in the limit of decreasing animal size for which the Rey- small animals for which urine flow is resisted by capillary and nolds number approaches zero. Because of isometry, V ∼ M and D ∼ viscous forces, which scale with the perimeter of the urethra. More M1/3.RewritingEq.7 at low Reynolds number, we have u ∼ D, accurate predictions for small animals require measurements of and therefore, the time to eject one drop from Eq. 10 scales as the urethral exit perimeter and the urethral cross-section, which is −1 −2/3 1/3 Tdrop ∼ Bo ∼ M .UsingEq.9, the final drop size is Rf ∼ D ∼ M . known to vary with distance down the urethra (67). Current By conservation of mass, a full bladder of volume V can produce models of noncircular pipe flow are not applicable, because they = = π 3 ∼ 2=3 N spherical drops, where N 3V 4 Rf M . Thus, the urina- only account for infinitesimal corrugations (68). Additional math- tion time for small animals T = NTdrop is constant and there- ematical techniques as well as accurate urethral measurements fore, independent of animal size. This prediction indicates are needed to increase correspondence with experiments. that small animals urinate for different durations than large animals, which is in correspondence with experiments. Our Materials and Methods experiments indicate that mammals of mass 0.03–0.3 kg uri- – We filmed urination of animals using both Sony HDR-XR200 and high-speed nate for durations of 0.1 2 s. We have insufficient range of cameras (Vision Research v210 and Miro 4). The masses of animals are taken masses for small animals to conclude our prediction that from annual veterinary procedures or measured using an analytical balance. urination time is constant in this regime. Flow rate Q is measured by simultaneous high-speed videography and The model yields insight into the challenges facedp byffiffiffi small manual urine collection using containers of appropriate size and shape. We animals. In Eq. 7, flow speed is positive only if Pbladder αD ≥ 4σ, σ use the open-source software Tracker to measure the time course of the where is the surface tension of urine, which for humans is com- radius of urine drops produced by rats and mice. parable with the surface tension of water (62). Thus, we predict thatp theffiffiffi smallest urethra to expel urine has a diameter of ACKNOWLEDGMENTS. We acknowledge photographer C. Hobbs and our 4σ= αPbladder ∼ 0:1 mm. According to our allometric trends, the hosts at Zoo Atlanta (R. Snyder), the University of Georgia (L. Elly), the smallest animal that can urinate independently corresponds to Atlanta Humane Society (A. Lopez), and the animal facilities at Georgia Tech a body mass of 0.8 g and urethral length of 1.7 mm. This mass (L. O’Farrell). We thank YouTube contributors, including Alex Cobb, Cole corresponds to that of altricial mice (0.5–3 g), which are de- Onyx, demondragon115, drakar2835, ElMachoPrieto83, Ilze Darguze, Joe pendent on their mother’s licking of excreted urine drops (63). BERGMANN, Joey Ponticello, krazyboy35, laupuihang, MegaTobi89, Mixetc, mpwhat, MrTitanReign, relacsed, RGarrido121, ronshausen63, Sandro Puelles, Discussion Silvia Lugli, and Tom Holloway. Our funding sources were National Science Foundation Faculty Early Career Development Program (Division of Physics) The urinary system works effectively across a range of length scales. Grant 1255127 for the modeling and Georgia Tech President’s Undergraduate This robustness is caused by the hydrodynamic contribution of the Research Awards for the experiments.

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